Video: Solving One-Step Linear Inequalities over the Set of Natural Numbers

Given that 𝑥 ∈ ℕ, determine the solution set of the inequality −𝑥 > −132.

02:13

Video Transcript

Given that 𝑥 is in the set of natural numbers, determine the solution set of the inequality negative 𝑥 is greater than negative 132.

We remember that this symbol that looks kind of like an N means natural numbers, which are positive integers. If 𝑥 is in the set of natural numbers, then it cannot be negative, nor can it be fractional. It must be a positive integer. If negative 𝑥 is greater than negative 132, how can we find 𝑥?

If we multiply negative 𝑥 by negative one, we would get 𝑥. But if we’re going to multiply or divide with inequalities, we need to remember that when we’re multiplying or dividing negatives, we must flip the sign. This means we would multiply both sides of the inequality by negative one. Negative 132 multiplied by negative one is 132. And then we would flip the inequality.

We now have something that says 𝑥 is less than 132. But we also know that 𝑥 is in the set of natural numbers. So first of all, that means we’ll need to use set notation, the curly brackets. And secondly, we’re only interested in the integers less than 132. 𝑥 can’t be negative, and it can’t be between whole numbers.

The smallest value 𝑥 could be would be zero. It would then be one, two, three, continuing. We can use an ellipse to represent that. And the highest value 𝑥 can be is 131. We need to be really careful here. Just because there’s 132 here doesn’t mean 𝑥 can be equal to 132. 𝑥 must be less than that. And so the largest integer that is less than 132 is 131. Under these conditions, 𝑥 can be all the positive integers between zero and 131.

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